Ben Smith scored a 100-ball 59 against his former club and down the order, Roger Sillence slammed an unbeaten 47-ball knock of 44 with three sixes as Worcestershire reached 191 for eight in 45 overs after a damp outfield resulted in delayed start.
Zaheer Khan struck a boundary in the only ball he faced.
Mongia trapped Garreth Batty for his only wicket of the match and he conceded 39 runs in his eight overs, which also included a maiden.
Chasing a victory target of 192, Leicestershire batsmen put up an abject show at home ground with the top five failing to reach the double digit.
Dinesh Mongia began with a four but in the first ball he faced, Zaheer induced him for an edge and the Punjab southpaw, in scintillating form so far, returned caught behind as wicketkeeper Steve Davies gleefully accepted the catch.
Earlier, Zaheer had claimed the important wicket of Darren Maddy (3) and his bowling figure read 6-0-22-0.
Paul Nixon's 95-ball knock of 67, with three boundaries and two sixes in it, went in vain as spinner Batty (4/27) ran through the middle and lower order and Leicestershire were bowled out for 130.
During his defiant knock, Nixon startled the opponents with two reverse sweeps for six off former Zimbabwe Test player Ray Price.
David Masters (12) was the only other Leicestershire batsman to reach double figures.
It was a depleted Leicestershire, which took the field yesterday without five key players, including skipper Jeremy Snape.
Brief score: Worcestershire 191 for 8 in 45 overs Leicestershire 130 all out in 39.1 overs